odel damped oscilliations using second order differential equations

Fundamental Concepts

Damped oscillations refer to oscillatory motion where the amplitude decreases over time.
Damped oscillations are represented mathematically using second order differential equations.
The general form of a damped oscillation is given by m * y’’ + b * y’ + k * y = 0, where y’‘ represents the second derivative of displacement, y’ represents the first derivative of displacement, m is the mass, b denotes the damping constant, and k denotes the stiffness.

Classification of Damping

Oscillations can be over-damped, critically damped or under-damped, depending on the value of the damping ratio.
The damping ratio is calculated by zeta = b / (2 * sqrt(m*k)).
If zeta < 1, the system is said to be under-damped. The system oscillates, but the amplitude of oscillation decreases over time.
If zeta = 1, the system is said to be critically damped. This represents the boundary between oscillatory and non-oscillatory systems.
If zeta > 1, the system is said to be over-damped. In this case, the system does not oscillate.

Calculating Essential Values

The natural frequency of undamped oscillation can be calculated using w_n = sqrt(k/m).
The damped frequency can be calculated using w_d = w_n * sqrt(1 - zeta^2). This applies only for under-damped systems.
The time constant of the system can be given by t = m/b, indicating the time it takes for the displacement to reduce to a certain fraction of its original value.

Solution of The Differential Equations

The nature of the differential equation’s solution is determined by the type of damping.
In the case of under-damped systems, the solution is a sum of an exponential decay and an oscillating function, given by y(t) = Aexp(-zetaw_nt)cos(w_d*t + phi).
In the case of critically damped systems, the solution includes an exponential decay then linear growth, given by y(t) = (A + Bt)exp(-zetaw_nt).
For over-damped systems, the solution is a sum of two exponential decays, given by y(t) = A* exp(r_1t) + B exp(r_2*t), where r1 and r2 are the roots of the characteristic equation.

End of revision for Damped Oscillations - remember to test your understanding by working through a variety of mathematical problems to consolidate these topics.